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Unformatted text preview: Phys. 7268 Assignment 2 Due: 2/10/11 Problem 1 Type of Linear Instability for a Swift-Hohenberg Equation with a Second-Order Time Derivative. Consider a two-dimensional Swift-Hohenberg equation with a second-order time derivative: ∂ 2 t u ( x,y,t ) = ru- (1 + ∇ 2 ) 2 u- u 3 . Show that the base solution u = 0 becomes linearly unstable at r = 0 and determine the type of instability ( I s , I o , III o , etc). Problem 2 Type of Instability for Two Coupled Reaction-Diffusion Equations. Consider the following reaction-diffusion evolution equations ∂ t u = au + ∂ 2 x u- b∂ 2 x v- ( u 2 + v 2 )( u + cv ) , ∂ t v = av + b∂ 2 x u + ∂ 2 x v- ( u 2 + v 2 )( v- cu ) for the scalar fields u ( x,t ) and v ( x,t ) in one dimension, where the parameters a , b , and c are arbitrary real constants. (You can think about the fields u and v as representing concentrations of two reacting and diffusing chemicals.) Derive a condition for the linear instability of the zero base solution ( u ,v ) = (0 , 0) and determine whether the instability is of type I s , I o , or III o . Hint: When you substitute the infinitesimal perturbation with the correct spatial and temporal dependence (the same for both fields) into the linearized PDEs, you will obtain an eigenvalue problem involving a...
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- Spring '11
- Derivative, Boundary value problem, Eigenvalue, eigenvector and eigenspace, Normal mode, Boundary conditions